UCSB DRP: Research on Analytic Number Theory (Riemann Sums, Mobius Function)

What is UCSB DRP?

The UCSB Directed Reading Program (DRP) is a selective mathematics mentoring program in which graduate students or junior faculty members mentor undergraduate students in reading and research projects.

Paired based on mutual interests, groups will then work together during the Winter and Spring quarters (16 weeks) to produce a poster to be presented to colleagues and those residing in the College of Letters and Sciences.

Please note that due to the pandemic, this experience was fully virtual.

More information can be found at the following link.

Purpose

This project explores the relationship between the Mobius Function, Mertens Function and their connections to the Riemann Hypothesis and Prime Number Theory in a 2-person team over 16 weeks.

The Process: A Brief Walkthrough the 16 Weeks

Early January

After accepting our invitations to program, students of similar interests (ours being Analytic Number Theory) are grouped together.

This was our first time meeting our mentor - David Nguyen, a 6th year PhD student and the head organizer of UCSB’s DRP and each other. Though we look super serious in the photo, the group meshed well! \[ \]

\[ \] We then decided to split into two groups so we could explore more topics - Tien and I (pictured on the right) became research partners and later good friends!

Late January

After first exploring on our own, then as a group of two, finally Tien, David and I decided to narrow our focus to the following topics: Riemann Sums and the Mobius Function.

We also choose our book, Graduate Texts in Mathematics: Problems in Analytic Number Theory, 2nd Edition by M. Ram Murty. (A phenomenal and digestible read!).

This would act as our guide in exploring topics, a place to practice problem solving and serve as a jumpstart to fostering questions we would like to answer in our own research.

\[ \] \[ \] \[ \] \[ \] \[ \] \[ \] \[ \]

Feburary to May

Tien, David and I then proceeded to meet once a week over the course of four months to discuss various things:

(Note: As this project had a lot of flexibility, Tien and I were also meeting on our own several times a week to discuss and work different parts of this project) \[ \] Getting to know each other and taking a look at David’s personal research. \[ \] \[ \] Discussing theory and techniques for solving different types of questions. \[ \] \[ \] Viewing examples in preparation to make our own poster. \[ \]

\[ \]

Python Code

Python code we used to reach conclusions seen in our presentation (graphs, square-free checks, etc.).

Utilizes NumPy and MatPlotLib, features unique functions:

Mobius Function.py

Produces the graph of \(P(x)\) vs \(x\) and conjectural bounds for \(\pi(x)\) - Fig.4 in the presentation \[ \]

main.py

As stated in the comments seen in the code, this program evaluates Mobius def (if M(N) is 1 - given N=1 or if M(N) is 0 - given any prime factor of N is contained twice. Also shows the case where M(N) = (-1)^(none distinct prime factors)) and isPrime def (checks if n is prime) \[ \]

base.py

Base case code supporting main.py. All prime factors are contained only once, return 1 if p is even, else return -1 \[ \]

mobuisgraph.py

Produces the graph of Mobius Function \(\mu(n)\) for \(n\le25\) - Fig.1 in the presentation \[ \]

sqroot.py

Produces the graph of former version of the Merten Conjecture (Disproven) - Fig.2 in the presentation \[ \]

Presentation

Due to COVID 19, the 2021 DRP poster viewing/presentation session was hosted on Gather-Town.

UCSB 2021 DRP Poster Session Event Program

My research partner, Tien and I are listed under poster session B group #4! \[ \] \[ \] Friends, family, groups and other math lovers waiting at the opening ceremony, located in the auditorium, with their custom avatars. Tien and I can be seen in the front row sitting together!
\[ \] \[ \] The empty poster presenting room where 16 groups will be presenting their topics of interest. \[ \]

Application of Prime Numbers Poster

\[ \] Tien and I set up at our station, ready to start presenting our topic - The Application of Prime Numbers; it’s a full house! \[ \] \[ \] Tien and I taking turns to present different sections of our poster and answering questions from our audience; the presentation was a success! \[ \]

Takeaway

From the 16 weeks, I personally developed these skills:

• The ability to produce fruitful results and successfully communicate my findings with little direction (this was a very open ended project)
• Thrive in both an independent and collaborative setting
• Set realistic deadlines to meet my own individual goals - would have a positive impact on our goals as a group
• Problem solving (again, the topic of number theory is so broad, often I would ask myself - how do we solve the unknown?)
• Utilizing documentation (Python documentation and from the analytic number theory reading)
• Being organized (this post and several memos!)
• Strengthen my knowledge of Python (NumPy and MatPlotLib)

Thanks for coming on this journey with me!